Given a simple noun phrase, we can again use state-of-the-art NLP annotation tools [1]} to parse that phrase into a dependency structure with head words \(u_h\) , prepositions \(u_{prep}\) , and objectives of preposition \(u_{pobj}\) . In navigation instructions, the preposition \(u_{prep}\)  normally specifies a spatial relationship between head word \(u_h\)  and the preposition objective \(u_{pobj}\) . In other words, \(u_{pobj}\)  serves as the dependency of \(u_{h}\)  and contributes to its unique grounding. Based on the above observation, we can first ground \(u_{pobj}\)  to the map \(m\)  and generate a belief of location reference, denoted as \(b_{pobj}\) . Then, we parse each element in the dependency structure in a bottom-up fashion until we reach the final destination.
The above grounding process can be written as
\(b_{pobj} &= f_\theta ^{h}(u_{pobj}, b_0)\\b_{prep} &= f_\theta ^{prep}(u_{prep}, b_{pobj})\\b_{h} &= f_\theta ^{h}(u_h, b_{prep}) \) 
